We present a fourth-order accurate, multilevel Maxwell solver, discretized
in space with a finite volume approach and advanced in time with the
classical fourth-order Runge Kutta method (RK4). Electric fields are
decomposed into divergence-free and curl-free parts; we solve for the
divergence-free parts of Faraday's Law and the Amp\`ere-Maxwell Law
while imposing Gauss' Laws as initial conditions. We employ a damping
scheme inspired by the Advanced Weather Research and Forecasting
Model to eliminate non-physical waves reflected off of coarse-fine
grid boundaries, and Kreiss-Oliger artificial dissipation to remove standing
wave instabilities. Surprisingly, artificial dissipation appears to damp the
spuriously reflected waves at least as effectively as the atmospheric
community's damping scheme.